Topology optimization of the flat steel shear wall based on the volume constraint and strain energy assumptions under the seismic loading conditions

In structural engineering systems, shear walls are two-dimensional vertical elements designed to endure lateral forces acting in-plane, most frequently seismic and wind loads. Shear walls come in a variety of materials and are typically found in high-rise structures. Because steel shear walls are lighter, more ductile, and stronger than other concrete shear walls, they are advised for usage in steel constructions. It is important to remember that the steel shear wall has an infill plate, which can be produced in a variety of forms. The critical zones in flat steel shear walls are the joints and corners where the infill plate and frame meet. The flat infill plate can be modified to improve the strength and weight performance of the steel shear walls. One of these procedures is Topology Optimization (TO) and this method can reduce the weight and also, increase the strength against the cyclic loading sequences. In the current research paper, the TO of the infill steel plate was considered based on the two methods of volume constraint and maximization of strain energy. Four different volumes (i.e., 60%, 70%, 80%, and 90%) were assumed for the mentioned element in the steel shear wall. The obtained results revealed that the topology configuration of CCSSW with 90% volume constraint presented the highest seismic loading performance. The cumulated energy for this type of SSW was around 700 kJ while it was around 600 kJ for other topology optimization configurations.

be made from a variety of materials, but the most common ones are plywood, steel, and concrete.Reinforced concrete is used in the building of shear walls to increase their stiffness and lateral strength.These concrete walls are most frequently used in medium to high-rise buildings 14 .Plywood shear walls, which are designed to withstand wind loads from hurricanes, tornadoes, and severe winds, are meant to be able to tolerate continuous lateral strains 15 .Steel plate shear walls are constructed using steel and infill plates.They are most commonly used in tall structures for seismic protection because they are made with boundary characteristics that can bear seismic stresses during events like earthquakes 16 .According to the points raised, steel shear walls are typically employed in steel constructions because they are lighter, more ductile, and stronger than other kinds of shear walls 17 .As previously mentioned, the steel shear walls consist of several components, such as columns, beams, and an infill plate.A key component in regulating the steel shear wall's seismic response is the infill plate 18 .
One of the key points in the field of strengthening the shear walls, is topology optimization.As a form optimization technique, topology optimization makes use of mathematical models to optimize material arrangement for a given set of loads, circumstances, and constraints inside a user-defined space 19 .By considering the topology optimization on the steel shear walls, the weight and strength can be reduced and improved, respectively.Therefore, topology optimization can be useful from a cyclic performance point of view.
In order to gain a better understanding of the seismic performances of shear walls, several researchers have recently attempted to investigate various techniques that may be effective against the seismic sequences.Some of the published articles about the aforementioned issues are included in the paragraph that follows.
The findings of Khan and Srivastava's 20 study on the inelastic behavior of steel plate shear walls under various opening conditions were published, and they showed that the opening systems had a major impact.Lastly, they offered novel models for the degradations of stiffness and shear strength.The greater length of the infill plate led to a better energy dissipation capability, and Mansouri et al. 21verified that the higher amounts of applied energy were dissipated through the infill plate.Using 57 computer models, Paslar et al. 22 investigated how the border elements and the infill plate of a steel plate shear wall are connected.Additionally, they used the experimental models to confirm the initial ones.In actuality, it was determined that a significant component was the way the infill plate and frame were connected.The obtained results showed that in comparison to the columnonly connected joint, the beam-only connected joint decreased the shear force resistive in the shear wall.The primary objective of the research has been to come up with novel approaches to enhance the steel plate shear walls' ductility, energy absorption, and other properties.For example, Lu et al. 23 suggested that the seismic loadresisting system in the steel structures be self-centering steel plate shear walls with slits.Finite element analysis proved the indicated shear walls' capabilities.When compared to traditional shear walls, Azandariani et al. 's 24 study showed the efficacy of the partial plate-column connection in steel plate shear walls.
Professional software is now made specifically for topology optimization thanks to the development of optimization techniques.Numerous research that optimizes the topology of structures has recently been reported.Topology optimization was utilized by Bagherinejad and Haghollahi 25 to determine the optimal form of perforated steel plate shear wall in the moment frames.They used the maximizing of reaction forces as the objective function and carried out the optimization.Daryan et al. 26 proposed an effective approach for the best design of frames with steel plate shear walls by utilizing bat optimization and modified dolphin echolocation techniques.
According to the above literature, using topology optimization to find the optimum shape of the infill plates of the steel shear walls can be useful in the presence of cyclic loading conditions.Thus, in this research article, three different connection types between the infill plate and the border elements (i.e., fully-connected, beamconnected, and column-connected) together with the two well-known methods of volume and strain energy were assumed for the analyses through the use of finite element analyses.Besides, four various volume constraints of 60%, 70%, 80%, and 90% were remarked for the infill plates of the steel shear walls.It is noteworthy that the selected assumption is considered for the first time and it can be stated that this paper considered comprehensive assumptions.Then, finite element analysis was implemented to figure out the optimum infill plate shapes and some critical characteristics of the steel shear walls including stress and strain distributions, hysteresis curves, strength, cumulated energy, and plastic dissipation energy.

Methodology
As stated earlier, the current research paper used the topology optimization concept in the steel shear walls to obtain the optimum shape of the infill plate of the steel shear wall by utilizing finite element simulations.Thus, first, this concept is briefly described and then the implemented experimental data together with the studied steel shear walls in the current research paper are stated.

Topology optimization (TO) methodology
The theory of TO is described here, along with the objective function, constraints, solution method, and general formulation.It should be noted that the Topology Optimization (TO) concepts are stated in this subsection to provide better clarifications about the TO's main formulations for the readers.However, TO procedures were carried out employing the Tosca plugin within Abaqus CAE.The volume constraints and strain energy minimization are the TO's objective function in this research.The energy bilinear form serves as the foundation for the formulation of TO employing gradient techniques such as sensitivity analysis 27 .Furthermore, it is feasible to perform a nonlinear FEA in a TO using the strain energy.A scalar parameter called strain energy (Π) represents the stored energy brought about by structural deformations.The energy is equivalent to the product of stress (σ) and strain (ε) on the element's volume (V).The sum of the strain energies in each element of a structure equals the total strain energy of the structure (see Eq. ( 1)).
where the element displacement vector (u e ) and the stiffness matrix (k e ), respectively, are shown.When a displacement pattern is applied to the structure as the loading, the strain energy should be maximized to maximize the structural stiffness 27 .60%, 70%, 80%, and 90% of the primary infill plate were the defined restrictions for the final volume of the optimized plate.To do the TO, there are three fundamental steps.Based on the findings of the finite element analysis, the first step is to determine the effective elements of the objective function.In the second stage, a penalization mechanism must be used to enhance and reduce, respectively, the density and stiffness of the effective and ineffective elements.All elements' density variables need to be adjusted in the third phase.Iteratively carrying out these actions is necessary until the stop criteria and the constraint are satisfied.Two well-known gradient methods are sensitivity analysis and optimality criteria.Here, the effective elements are found using sensitivity analysis, the density of the elements is changed using Simple Isotropic Material with Penalization (SIMP), and the design variables are updated using the Method of Moving Asymptotes (MMA).The SIMP-based TO formulation can be expressed as follows: where F is the prescribed volume fraction (f = 0.50), Ψ is the objective function, ρ e is the density of the elements (the design variable), p is the penalization power (p = 3), V (x) and V o are the material volume and design domain volume, respectively, K is the global stiffness matrix, ρ min is the minimum relative density (ρ min = 0.001), and N is the number of elements used to discretize the design domain.According to 27 , p = 3 is an appropriate value based on the numerical results of completed experiments.The following could be used to express how sensitive strain energy (the objective function) is to element density (the design variable): Taking into account the generic static equation (KU = F): Using Eq. ( 5) in place with Eq. ( 4): The following is one possible way to write Eq. ( 6) using the SIMP approach and Eq. ( 2): The solution method and mechanism for updating the element density in each iteration is the Method of Moving Asymptotes (MMA).By using Eq. ( 8), MMA approximates a function (Ψ) of n real variables x = (x 1 .… .xn ) around a given iteration point x 0 .
where the top and lower vertical asymptotes for the approximations of Ψ are denoted by U i and L i , respectively.The sensitivity of strain energy is shown to be negative for all element densities by Eq. ( 7).Thus, following iteration j, the MMA formulation (Eq.( 8)) could be expressed as Eq. 9 27 . (1) u T e k e u e (2) max : (ρ e ) P u T e k e u e (3) Subject to : where υ e is an element's volume and L e is the MMA method's lower bound that could be modified in each iteration.The number of iterations that equals 100 is the initial stop condition.Additionally, two stop conditions are offered in the TO in order to avoid performing additional iterations.According to Eq. ( 10), these two stop criteria comprise the average changes of element density (l 2 ) and the rate changes of the objective function (l 1 ).
Figure 1 presents a flowchart of the TO processes based on the SIMP, MMA, and sensitivity analysis approaches.

Experimental data
Here, the reported results by Emami et al. 28 were considered as the main input data for the analyses.They tested three different types of shear walls; flat Steel Shear Walls (SSWs), vertically Corrugated Steel Shear Walls (CSSWs), and horizontal ones.The results revealed that the corrugated shear walls could present higher cyclic performance compared to the flat SSW.But from an economic issues point of view, the flat SSW was a good choice.The schematic representation and the real test specimen of the flat SSW are exhibited in Fig. 2.
In Fig. 2a, some technical codes are reported which relate to the technical information about the construction of the flat SSW.These technical data are gathered in Table 1.
The main concepts and reported results of the current research paper rely on the basis of the flat SSW.The objectives of this research study are noted in the next subsection.

Studied cases and purposes
As stated earlier, the steel shear walls play a vital role in controlling the damping the seismic loading sequences in the steel structures.It should be noted that in the steel shear walls, the infill plate is the major effective element among the other elements naming beam and column.Different shapes are recommended for the infill plate based on the literature review 29 but still, more endeavors are needed to improve the infill plate's performance against the cyclic loading conditions.For instance, some researchers proposed corrugated steel shear walls with different corrugation shapes 30 .Here, the current research paper used Topology Optimization (TO) to obtain the  optimum shape of the infill steel plate.To optimize the infill plate according to the TO theories, four different volumes of 60%, 70%, 80%, and 90% were stipulated.For clarification, the 60% meant that the optimized infill plate had 60% volume of the initial infill plate.Another constraint that was considered was the minimization of the strain energy in the infill plates of the SSW during the cyclic loading conditions.This assumption resulted in lowering the amounts of the stresses in the concentration zones.
Besides the mentioned consideration above, three other assumptions were also considered in the TO theories.Here, different zones in the infill plate were considered frozen areas, which meant that these areas were out of the optimization scope.The frozen areas and other areas as the targeted zones of the optimization are illustrated in Fig. 3.
After considering the mentioned assumptions, the TO theories together with the finite element analysis software were implemented to optimize the infill plate of the SSW.The procedure of the finite element analysis is briefly described in the next subsection.
To clarify the naming system in the current research paper, the nomenclature of the samples for the finite element analysis and TO results are reported in Table 2.  Finite Element Analysis (FEA) is the process of modeling objects and systems in a computerized environment with the objective of locating and fixing potential (or existing) structural or performance issues 31 .Preprocessing, which gets the modeling data, processing, which puts the equations together and solves them, and postprocessing, which shows the analysis results, are the three primary steps in the finite element analysis process 32 .The automotive, aerospace, shipbuilding, and construction industries can all benefit greatly from the use of FEA 33 .It provides precise, efficient, and economical answers to engineering problems.
In the current research paper, the SSWs were modeled in the FEA software (Abaqus CAE) and then, the mechanical properties according to Table 3 were assumed as the material properties of the elements.After assembling the elements together, three different solvers were used; nonlinear static and dynamic explicit.It is noteworthy that in the current research paper, to obtain the optimized shape of the infill plate, the TOSCA 34 plugin was used.Where the mentioned solvers were implemented to obtain the pushover curves, and hysteresis behavior, respectively.The loading and boundary conditions were applied based on Fig. 4 in the optimized SSWs.
The loading sequences were applied to the optimized SSWs according to Fig. 5.This loading condition presents seismic loading sequences like earthquakes.
The designed shear walls were meshed apart following the aforementioned processes (see Fig. 6).Also, after performing the TO procedures and implementing the mentioned plugin in the FEA, the optimized SSWs were also meshed apart.Figure 7 shows a representative illustration of the SSW with an optimized infill plate.The mesh convergence analysis was also used to determine the appropriate number and sizes of elements for the shear walls, with the element type being S8R that is based on the reduced integration method and with the order of two.In the end, a 5 cm mesh size was used for the frame parts and a 2.5 cm mesh size for the infill plate.

Validation study
Any FEA model initially should be verified by some experimental results.This issue confirms the applicability of the modeled samples.The modeled samples in the FEA section of this paper were validated by the results reported by Emami et al. 28 .For this aim, the hysteresis and pushover curves that were obtained numerically here were compared to those reported by the experimental study 28 .The hysteresis behavior and pushover performance of the flat SSW are presented in Figs. 8 and 9, respectively.Based on these figures, the modeled samples in the current research paper could predict the SSWs' performance with minor errors.

Recommended infill plates (optimized)
According to Fig. 2, three different conditions were assumed for the topology optimization besides, four volume constraints of 60%, 70%, 80%, and 90% together with the minimization of the strain energy were considered.
Based on the stated approaches in section "Methodology" and FEA procedures the recommended and optimized

Stress distribution
The von-Mises stress distributions in the different configurations of the SSWs are illustrated in this subsection.Figure 13 shows the stress distribution in the SSW with the flat infill plate.Based on this figure, the corners where the infill plate joined to the other elements resulted in higher amounts of stress.Thus, these zones are known as the stress concentration zones.Figures 14, 15, and 16 present the stress distributions in the FCSSW,    BCSSW, and CCSSW configurations, respectively.The TO procedure could reduce the amounts of stress in the stress concentration zones and also, helped to good distribution of the stress over the infill plate of the SSWs.This claim was completely approved at the bottom of the columns of the optimized SSWs in Figs. 14 and 15.

Plastic strain distribution
Besides the stress distribution, the plastic strain distribution obtained from the FEA can be helpful because in most cases, higher plastic strains can result in structure collapsing.To this end, the plastic distributions in different SSWs are illustrated in the current section.Figure 17 shows the plastic strain values and their distributions in the SSW with a fully flat infill plate.Based on this figure, the corners of the SSW experienced higher plastic strains compared to the other vicinities in the SSW.However, using topology optimization resulted in distributing the plastic strains all over the infill plate.Figures 18, 19, and 20 relate to the SSWs with configurations of FCSSW, BCSSW, and CCSSW, respectively.Two configurations of FCSSW and BCSSW could reduce the amounts of plastic distributions in the SSW and the lowest amount related to the 80% volume  www.nature.com/scientificreports/constraint.The CCSSW configuration (based on Fig. 20) increased the plastic strain distributions' amounts, which meant that these SSWs experienced more plastic strains and consequently more plastic deformation.Thus, the CCSSW could absorb more exerted energies by utilizing plastic deformations.One of the consequences of the mentioned issue was the out-of-plane displacements which are discussed extensively in the next subsection.

Out-of-plane displacement
One of the most important challenges that the shear walls face is the high out-of-plane displacement.Thus, controlling this issue can be helpful to the seismic performance of the shear walls.The out-of-plane distributions that were evaluated from the FEA are presented in Figs.21, 22, 23, and 24 for the SSWs' configurations of flat infill plate, FCSSW, BCSSW, and CCSSW, respectively.Comparing the obtained results in these figures revealed that the highest values of out-of-plane displacement related to the fully flat infill plate SSW (see Fig. 21).Implementing the TO could reduce these values, thus, the optimized SSW experienced lower out-of-plane displacement.Hence, the optimized SSW could be more applicable and safer in comparison to the simple SSW from a seismic loading condition point of view.Considering the volume constrained that were assumed in the TO procedure, the 80% and 90% volume constraints could present lower out-of-plane displacement, thus, these two constraints could be more reliable than the other two volume constraints.
It should be noted that in some cases, the 60% and 70% volume constraints resulted in higher out-of-plane displacements compared to the flat SSW that is exhibited in Fig. 21.

Hysteresis curves
When a loading sequence is applied to a structure cyclically, the mentioned structure experiences the hysteresis behavior.In the current research paper, the seismic loading condition was applied according to the curve plotted in Fig. 5, and then, the hysteresis responses of the SSWs were recorded in the FEA and plotted in a single curve.
As stated earlier in the FEA section, the loading condition that was applied to the SSWs was displacement control, thus, the hysteresis curves presented constant values of displacement but the evaluated forces were different.The  Comparing the presented results gave a better inside into the seismic performance of the various types of studied SSWs.In each category of the SSW, the 80% and 90% volume constraints especially the 90% could result in the higher area inside the hysteresis loop in comparison to the other ones.As a matter of fact, the 90% volume constraint was similar to the 100% one (the fully flat SSW that was exhibited in Fig. 8) but the only problem with this volume constraint was the higher weight and costs of fabrication processes.However, in a single case of volume constraint (e.g., 90%) the CCSSW configuration of SSW presented the best response from a hysteresis curve point of view.The main reason for this issue was the frozen area in the TO procedure because, in a shear wall, the columns play a critical role in controlling the structure strength.The other parameter, strength, computed and plotted in a curve with the variation in drift angle (i.e., Strength-Drift Angle curves) after the hysteresis curves of the various shear walls were recorded.Each hysteresis curve's peak loads were chosen, combined, and given a strength designation.Additionally, the drift angle was named after dividing the horizontal displacement by the shear wall column's length.Plotted strength-drift angle curves for various shear wall configurations, including the Simple shear wall, are displayed here.The mentioned results are plotted in the shown curves of Figs. 28, 29, and 30.These curves also confirmed that the volume constraint of 90% had the highest value of strength and also, the CCSSW configuration of SSW was the best optimized SSW compared to the other two systems.

Cumulated energy
Each structure when exposes to the cyclic loading conditions, in each cycle, some exerted energy cumulates in the structure.As the structure absorbs more energy (i.e., cumulated one) means that the structure has a great performance against the seismic loads.Cumulated energy as a response to the SSWs was obtained from the FEA and presented in Figs.31, 32, and 33 for configurations of FCSSW, BCSSW, and CCSSW, respectively.
Two important results were drawn from this subsection; first, the 90% volume constraint could result in higher amounts of cumulated energy, second, the CCSSW configuration was the best configuration among all three types of shear walls.Again, the frozen area in the CCSSW could resist the seismic loading sequences and absorb higher amounts of exerted energy.
For more detailed information, suppose a 60% volume constraint, all three types of TOs presented cumulated energy of around 600 kJ.Considering the FCSSW and BCSSW TOs resulted in cumulated energy of around 600 kJ but this value was increased up to 700 kJ for the CCSSW TOs.This issue was also repeated for other volume constraints.Thus, the CCSSW configuration presented good performance from a cumulated energy point of view.To conclude about obtained results, the plastic dissipation energies were calculated and plotted in the next subsection.

Plastic dissipation energy
One of the most important aspects of SSWs is the high ductility during the loading conditions.The steel material presents ductile behavior due to its strain-hardening phenomenon.Thus, the SSWs show ductile behavior during earthquakes.It should be noted that any exerted load should be spent on a material and structure.There are two ways; absorbing exerted energy by producing the plastic zones and creating and expanding the cracks in a structure.Therefore, if a structure presents ductile behavior and consequently plastic deformation, the failure will be postponed.was due to the lower amounts of used material in this configuration.There was not enough steel material to absorb the exerted energy.
Another important result that was obtained here was the great performance of the CCSSW types from a plastic dissipation energy point of view.It is noteworthy that the main reason for these results was that the columns in this type were considered frozen in the TO procedure and the columns are the most important elements in the civil structures after the beams.

Conclusions
The flat Steel Shear Walls (SSWs) are the common elements used in steel structures for damping the loads exerted by cyclic loading conditions like earthquakes.One of the important parts of the SSWs is the infill plate.In the current research paper, the Topology Optimization (TO) was used to optimize the shape of the steel infill plate of the SSWs.Two assumptions of strain energy and volume constraint were applied in the TO procedures for finding 14:10323 | https://doi.org/10.1038/s41598-024-61204-1

Figure 2 .
Figure 2. (a) schematic illustration and (b) real tested specimen of the fabricated flat steel shear wall 28 .

Figure 3 .
Figure 3. Different zones in the optimization process of the SSWs.

Figure 5 .
Figure 5. Loading sequences that were applied to the optimized SSWs in the FEA 28 .

Figure 6 .
Figure 6.Mesh pattern in the SSW with the flat infill plate.

Figure 7 .
Figure 7. Mesh pattern in the SSW with an optimized infill plate.

Figure 8 .
Figure 8. Hysteresis curves of the flat SSW that were obtained numerically and experimentally 28 .

Figure 9 .
Figure 9.Comparison of the pushover curves that were evaluated experimentally 28 and numerically.

Figure 13 .
Figure 13.Stress distribution in the flat SSW.

Figure 17 .
Figure 17.Plastic strain distribution in the SSW with the flat infill plate.

Figure 21 .
Figure 21.Out-of-plane displacement distribution in the SSW with the flat infill plate.

Figure 29 .
Figure 29.Strength-drift angle for the BCSSW configuration considering various volume constraints.

Figure 36 .
Figure 36.Plastic dissipation energy of CCSSW configuration with different volume constraints.

Table 1 .
28chnical data of the flat SSW reported by Emami et al.28.

Table 2 .
Nomenclature of the samples.

Table 3 .
Material properties for three elements in the SSW for the FEA.